PromptGen with conditional INN   
To generalize the control to continuous values, e.g., the scene parameters \(\mathbf {\rho }\)  in Eq. (REF ), we condition the INN on \(\mathbf {\rho }\) .
The condition is modeled by replacing \(\mathbf {z}= f_{\mathbf {\theta }}(\mathbf {\epsilon })\)  with \(\mathbf {z}= f_{\mathbf {\theta }}(\mathbf {\epsilon }, \mathbf {\rho })\) . Space limited, we provide details and derivations in Appendix REF .
This extension results in a similar architecture to StyleFlow [1]}, but StyleFlow [1]} uses MLE training on image-label pairs and is only applicable to explicit condition, which is not capable of modeling EBMs.
When the condition is modeled by the equivariant operations proposed by [3]}, PromptGen also satisfies the homomorphism property [3]} in the latent space.
